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Properties

Label 1-1407-1407.5-r1-0-0
Degree $1$
Conductor $1407$
Sign $0.938 + 0.346i$
Analytic cond. $151.203$
Root an. cond. $151.203$
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  + (−0.786 + 0.618i)2-s + (0.235 − 0.971i)4-s + (0.888 + 0.458i)5-s + (0.415 + 0.909i)8-s + (−0.981 + 0.189i)10-s + (0.0475 + 0.998i)11-s + (0.415 − 0.909i)13-s + (−0.888 − 0.458i)16-s + (0.723 − 0.690i)17-s + (−0.928 + 0.371i)19-s + (0.654 − 0.755i)20-s + (−0.654 − 0.755i)22-s + (−0.981 − 0.189i)23-s + (0.580 + 0.814i)25-s + (0.235 + 0.971i)26-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (0.235 − 0.971i)4-s + (0.888 + 0.458i)5-s + (0.415 + 0.909i)8-s + (−0.981 + 0.189i)10-s + (0.0475 + 0.998i)11-s + (0.415 − 0.909i)13-s + (−0.888 − 0.458i)16-s + (0.723 − 0.690i)17-s + (−0.928 + 0.371i)19-s + (0.654 − 0.755i)20-s + (−0.654 − 0.755i)22-s + (−0.981 − 0.189i)23-s + (0.580 + 0.814i)25-s + (0.235 + 0.971i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.938 + 0.346i$
Analytic conductor: \(151.203\)
Root analytic conductor: \(151.203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1407,\ (1:\ ),\ 0.938 + 0.346i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.602582924 + 0.2861456811i\)
\(L(\frac12)\) \(\approx\) \(1.602582924 + 0.2861456811i\)
\(L(1)\) \(\approx\) \(0.8568073727 + 0.2301328955i\)
\(L(1)\) \(\approx\) \(0.8568073727 + 0.2301328955i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.786 + 0.618i)T \)
5 \( 1 + (0.888 + 0.458i)T \)
11 \( 1 + (0.0475 + 0.998i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (0.723 - 0.690i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
23 \( 1 + (-0.981 - 0.189i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.995 - 0.0950i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (0.981 + 0.189i)T \)
53 \( 1 + (0.235 - 0.971i)T \)
59 \( 1 + (-0.995 - 0.0950i)T \)
61 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (0.888 - 0.458i)T \)
79 \( 1 + (0.995 - 0.0950i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
97 \( 1 + T \)
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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

−20.59679367991934503085295544639, −19.74883591904962544748939361662, −18.939434059130420405877938853407, −18.461300151462804336876420806549, −17.53081458253289315140505440439, −16.76925966678960588548525237967, −16.50659127438391446619634932850, −15.45886912937851442757097985342, −14.17600059375291698598204835480, −13.61680593530890420925551552461, −12.74555159968729656833157778268, −12.11145216267052474141163596435, −11.056008808319281300435658450835, −10.57836876544670031990313017129, −9.589517056088157471704546501922, −8.951941129105682970084136597931, −8.39181812067236113994144199309, −7.37795532601780808058463952444, −6.281359335665331408451712660390, −5.68137625167911617504005997062, −4.28290085882244857086745711946, −3.565766556032338324662285579575, −2.3485071922715851297693149255, −1.654857424506076519774679826513, −0.728652955746819559895918528, 0.54478997904423008080755006712, 1.78748121729761114770238158448, 2.37801802708750847859842670053, 3.75146758394735265200443907662, 5.07062574288383714423752543721, 5.77312969575087126845306815126, 6.44306509888241017058088975922, 7.42588764346941417959911578687, 7.93644474471113823169275405393, 9.19675571437691032151989685493, 9.59820842516174899986428281508, 10.53298353936131766249778341014, 10.907899856579337552091043504862, 12.26072398933105849441419301348, 13.060807761527361666649788123563, 14.13226740390189569089892523874, 14.59656139271614648006980514792, 15.34883120164141685934555030535, 16.20319577954280308238108861976, 17.00386344144572094512940979522, 17.69978749685264753120076028611, 18.22093619561249173226903342234, 18.82041157138864862332663462087, 19.840908985077657105416894055982, 20.554822709052748851475470577717

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