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Properties

Label 2-700-100.23-c1-0-41
Degree $2$
Conductor $700$
Sign $0.880 - 0.474i$
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  + (0.518 + 1.31i)2-s + (−0.799 − 1.56i)3-s + (−1.46 + 1.36i)4-s + (2.23 − 0.0995i)5-s + (1.64 − 1.86i)6-s + (0.707 + 0.707i)7-s + (−2.55 − 1.21i)8-s + (−0.0594 + 0.0817i)9-s + (1.28 + 2.88i)10-s + (0.257 + 0.354i)11-s + (3.30 + 1.20i)12-s + (−0.0479 − 0.302i)13-s + (−0.563 + 1.29i)14-s + (−1.94 − 3.42i)15-s + (0.276 − 3.99i)16-s + (4.64 + 2.36i)17-s + ⋯
L(s)  = 1  + (0.366 + 0.930i)2-s + (−0.461 − 0.905i)3-s + (−0.731 + 0.682i)4-s + (0.999 − 0.0445i)5-s + (0.673 − 0.761i)6-s + (0.267 + 0.267i)7-s + (−0.902 − 0.430i)8-s + (−0.0198 + 0.0272i)9-s + (0.407 + 0.913i)10-s + (0.0777 + 0.106i)11-s + (0.955 + 0.347i)12-s + (−0.0132 − 0.0839i)13-s + (−0.150 + 0.346i)14-s + (−0.501 − 0.884i)15-s + (0.0692 − 0.997i)16-s + (1.12 + 0.574i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.880 - 0.474i$
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: \((2,\ 700,\ (\ :1/2),\ 0.880 - 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76362 + 0.444919i\)
\(L(\frac12)\) \(\approx\) \(1.76362 + 0.444919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.518 - 1.31i)T \)
5 \( 1 + (-2.23 + 0.0995i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.799 + 1.56i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-0.257 - 0.354i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.0479 + 0.302i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-4.64 - 2.36i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.262 - 0.806i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.747 + 4.72i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-4.75 - 1.54i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.63 + 1.18i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.47 + 0.708i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-2.57 - 1.87i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-0.555 + 0.555i)T - 43iT^{2} \)
47 \( 1 + (4.58 - 2.33i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (2.52 - 1.28i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-9.05 - 6.58i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (5.21 - 3.79i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.746 - 1.46i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-4.33 - 1.40i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.20 + 1.14i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-2.03 + 6.26i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.813 + 0.414i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (5.82 + 8.01i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (4.98 + 9.78i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.35445172308670753572970478055, −9.541622420526063423134607529863, −8.540720634600651442326216486871, −7.73858550177232379513635316291, −6.75832335346467463530710072506, −6.12164019472649337391928683883, −5.47916861660107857775078546894, −4.36563454266260757344448694398, −2.79710914539569981104049539329, −1.19978513229054122181265352400, 1.27854024345740308939161884728, 2.70154069269672276717980655607, 3.87034589111929079506217196564, 4.97304400020653473669001797613, 5.40582989358178911085626842371, 6.48953795474645559946014608918, 7.952260936910546678663456120197, 9.238453315810164938765404109449, 9.810301015755207111445932361503, 10.32833845925042894798880564771

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