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Properties

Label 2-690-15.2-c1-0-19
Degree $2$
Conductor $690$
Sign $0.113 + 0.993i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.639 − 1.60i)3-s − 1.00i·4-s + (−1.63 + 1.52i)5-s + (1.59 + 0.685i)6-s + (0.772 + 0.772i)7-s + (0.707 + 0.707i)8-s + (−2.18 + 2.05i)9-s + (0.0731 − 2.23i)10-s − 1.36i·11-s + (−1.60 + 0.639i)12-s + (0.808 − 0.808i)13-s − 1.09·14-s + (3.50 + 1.64i)15-s − 1.00·16-s + (2.66 − 2.66i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.369 − 0.929i)3-s − 0.500i·4-s + (−0.729 + 0.683i)5-s + (0.649 + 0.279i)6-s + (0.291 + 0.291i)7-s + (0.250 + 0.250i)8-s + (−0.727 + 0.686i)9-s + (0.0231 − 0.706i)10-s − 0.411i·11-s + (−0.464 + 0.184i)12-s + (0.224 − 0.224i)13-s − 0.291·14-s + (0.904 + 0.425i)15-s − 0.250·16-s + (0.645 − 0.645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.113 + 0.993i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.113 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.476778 - 0.425412i\)
\(L(\frac12)\) \(\approx\) \(0.476778 - 0.425412i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.639 + 1.60i)T \)
5 \( 1 + (1.63 - 1.52i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-0.772 - 0.772i)T + 7iT^{2} \)
11 \( 1 + 1.36iT - 11T^{2} \)
13 \( 1 + (-0.808 + 0.808i)T - 13iT^{2} \)
17 \( 1 + (-2.66 + 2.66i)T - 17iT^{2} \)
19 \( 1 + 1.89iT - 19T^{2} \)
29 \( 1 + 1.81T + 29T^{2} \)
31 \( 1 + 5.22T + 31T^{2} \)
37 \( 1 + (7.05 + 7.05i)T + 37iT^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 + (-4.99 + 4.99i)T - 43iT^{2} \)
47 \( 1 + (-7.28 + 7.28i)T - 47iT^{2} \)
53 \( 1 + (1.52 + 1.52i)T + 53iT^{2} \)
59 \( 1 + 5.33T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + (-8.07 - 8.07i)T + 67iT^{2} \)
71 \( 1 + 2.85iT - 71T^{2} \)
73 \( 1 + (-8.07 + 8.07i)T - 73iT^{2} \)
79 \( 1 - 4.48iT - 79T^{2} \)
83 \( 1 + (0.994 + 0.994i)T + 83iT^{2} \)
89 \( 1 - 3.62T + 89T^{2} \)
97 \( 1 + (0.314 + 0.314i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.54345270083642028218091534139, −9.108147481358874888586334327196, −8.372363710308280853783012816366, −7.39231951622981649997396089403, −7.10858567416584979917839463058, −5.91832227403715866466084854497, −5.20216538461062318914065986252, −3.53414064219215519393132507380, −2.17520055538799613286074928062, −0.46062905904721099670482659475, 1.31940135323101332571793018491, 3.26805787700089153981497357521, 4.15520632459594383439515536723, 4.90811699303929699237056479383, 6.10190997745990794846395543620, 7.49350464807375410502480202316, 8.250595105080702077942228019305, 9.120335770894043381832373513796, 9.795454322945497986340772243899, 10.78384627907827207304008684581

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