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Properties

Label 2-700-100.23-c1-0-40
Degree $2$
Conductor $700$
Sign $-0.904 - 0.426i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
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  + (0.817 + 1.15i)2-s + (1.44 + 2.82i)3-s + (−0.662 + 1.88i)4-s + (1.37 − 1.76i)5-s + (−2.08 + 3.97i)6-s + (0.707 + 0.707i)7-s + (−2.71 + 0.778i)8-s + (−4.15 + 5.71i)9-s + (3.15 + 0.149i)10-s + (2.81 + 3.87i)11-s + (−6.28 + 0.844i)12-s + (−1.01 − 6.40i)13-s + (−0.237 + 1.39i)14-s + (6.96 + 1.35i)15-s + (−3.12 − 2.50i)16-s + (1.84 + 0.939i)17-s + ⋯
L(s)  = 1  + (0.578 + 0.815i)2-s + (0.831 + 1.63i)3-s + (−0.331 + 0.943i)4-s + (0.616 − 0.787i)5-s + (−0.850 + 1.62i)6-s + (0.267 + 0.267i)7-s + (−0.961 + 0.275i)8-s + (−1.38 + 1.90i)9-s + (0.998 + 0.0473i)10-s + (0.848 + 1.16i)11-s + (−1.81 + 0.243i)12-s + (−0.281 − 1.77i)13-s + (−0.0635 + 0.372i)14-s + (1.79 + 0.350i)15-s + (−0.780 − 0.625i)16-s + (0.447 + 0.227i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.904 - 0.426i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.610192 + 2.72701i\)
\(L(\frac12)\) \(\approx\) \(0.610192 + 2.72701i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.817 - 1.15i)T \)
5 \( 1 + (-1.37 + 1.76i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.44 - 2.82i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-2.81 - 3.87i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.01 + 6.40i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-1.84 - 0.939i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (1.91 + 5.89i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.775 - 4.89i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-1.53 - 0.498i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.93 + 1.27i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.987 - 0.156i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-1.40 - 1.01i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (1.72 - 1.72i)T - 43iT^{2} \)
47 \( 1 + (-4.01 + 2.04i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-5.09 + 2.59i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (10.8 + 7.91i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.26 - 3.10i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.64 + 5.19i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (1.52 + 0.496i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.38 + 0.377i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-0.0968 + 0.298i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.91 - 1.48i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (4.94 + 6.79i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.74 - 9.30i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.46551919229905605637660470748, −9.672784619908177694816597864850, −9.110704544596295413256170372954, −8.354794901044074990414987392846, −7.56322754043472502283155922688, −6.02814824616751756083928746703, −5.02741921222252379929805641849, −4.70071437266132738061091971353, −3.59053596656588737281129202171, −2.50146077889762662848089136031, 1.27811979836295040591421993174, 2.10871814429057263054253311315, 3.09020693573234081590217990630, 4.11616460622893064070169116166, 5.99301340114312199859053897418, 6.40426607101377864642303651120, 7.24371043116575980191906489738, 8.506014516805305282643845133014, 9.132003302707369260649436885386, 10.16795703359073092689547889688

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