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Properties

Label 1-967-967.155-r1-0-0
Degree $1$
Conductor $967$
Sign $0.931 + 0.362i$
Analytic cond. $103.918$
Root an. cond. $103.918$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.505 + 0.862i)2-s + (0.945 − 0.325i)3-s + (−0.488 + 0.872i)4-s + (0.911 + 0.410i)5-s + (0.759 + 0.651i)6-s + (−0.998 − 0.0520i)7-s + (−0.999 + 0.0195i)8-s + (0.787 − 0.615i)9-s + (0.107 + 0.994i)10-s + (0.00975 − 0.999i)11-s + (−0.177 + 0.984i)12-s + (−0.861 + 0.508i)13-s + (−0.460 − 0.887i)14-s + (0.995 + 0.0909i)15-s + (−0.522 − 0.852i)16-s + (−0.998 − 0.0585i)17-s + ⋯
L(s)  = 1  + (0.505 + 0.862i)2-s + (0.945 − 0.325i)3-s + (−0.488 + 0.872i)4-s + (0.911 + 0.410i)5-s + (0.759 + 0.651i)6-s + (−0.998 − 0.0520i)7-s + (−0.999 + 0.0195i)8-s + (0.787 − 0.615i)9-s + (0.107 + 0.994i)10-s + (0.00975 − 0.999i)11-s + (−0.177 + 0.984i)12-s + (−0.861 + 0.508i)13-s + (−0.460 − 0.887i)14-s + (0.995 + 0.0909i)15-s + (−0.522 − 0.852i)16-s + (−0.998 − 0.0585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.931 + 0.362i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.931 + 0.362i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.645945137 + 0.6850513688i\)
\(L(\frac12)\) \(\approx\) \(3.645945137 + 0.6850513688i\)
\(L(1)\) \(\approx\) \(1.729663636 + 0.5792000775i\)
\(L(1)\) \(\approx\) \(1.729663636 + 0.5792000775i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.505 + 0.862i)T \)
3 \( 1 + (0.945 - 0.325i)T \)
5 \( 1 + (0.911 + 0.410i)T \)
7 \( 1 + (-0.998 - 0.0520i)T \)
11 \( 1 + (0.00975 - 0.999i)T \)
13 \( 1 + (-0.861 + 0.508i)T \)
17 \( 1 + (-0.998 - 0.0585i)T \)
19 \( 1 + (-0.254 - 0.967i)T \)
23 \( 1 + (0.724 - 0.689i)T \)
29 \( 1 + (0.977 + 0.212i)T \)
31 \( 1 + (-0.347 + 0.937i)T \)
37 \( 1 + (0.996 - 0.0844i)T \)
41 \( 1 + (0.917 + 0.398i)T \)
43 \( 1 + (0.971 + 0.238i)T \)
47 \( 1 + (0.936 - 0.350i)T \)
53 \( 1 + (0.613 - 0.789i)T \)
59 \( 1 + (-0.209 - 0.977i)T \)
61 \( 1 + (0.113 + 0.993i)T \)
67 \( 1 + (0.638 - 0.769i)T \)
71 \( 1 + (0.494 + 0.869i)T \)
73 \( 1 + (0.613 + 0.789i)T \)
79 \( 1 + (0.906 - 0.422i)T \)
83 \( 1 + (0.0357 - 0.999i)T \)
89 \( 1 + (-0.985 - 0.168i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.37175443326598259778563488504, −20.72117791761204305783303815879, −19.97798999392085825882794395140, −19.56047377201156178824044465620, −18.62945112625592288741516109492, −17.72245269088569933583826287101, −16.82311715516375726029800925783, −15.58991255457379254869440916156, −15.061117502247199127632481181123, −14.15410599890325091614560680048, −13.43580637601617634919135780167, −12.731304009558666375202530704637, −12.344542178825345286232117650802, −10.75298135844526085884993410162, −10.05743064229217676642944440436, −9.45762475680188215516949292237, −9.017630672445381463354938525626, −7.65392621361207816128461861516, −6.49372130170812412531947546852, −5.513268427098450226311457312924, −4.5603765653917189037920232048, −3.83501798028675208141564819602, −2.47585685515157721197507288401, −2.37197743267251673914297325009, −0.98565400367919855657566508856, 0.63096269604600720666382374368, 2.55150491266161477005667897803, 2.783104336181956589328393517947, 3.95912291025569327044207481871, 4.99010362488693832204518644510, 6.26040504005983893141549820809, 6.68971965451682628233764953105, 7.36745518376803150888731968908, 8.75860857606182716392495519445, 9.041719806891707121576596032265, 9.9587781716741425617598100867, 11.164621716483534429999832584415, 12.60709934008157930173619810356, 13.00205516369371357696791482444, 13.844737667645659988400268572219, 14.25532862043594804229214473196, 15.142109433111799024341752453522, 15.946743529671629439453465001786, 16.74478290051019178918083898855, 17.636819745011431594073419640713, 18.38792360066733440195797088335, 19.24101947277911930965491756910, 19.90568771103670915265592755684, 21.16004438507577318835033615506, 21.720819938553140031746432137354

Graph of the $Z$-function along the critical line