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Properties

Label 2-700-140.19-c1-0-26
Degree $2$
Conductor $700$
Sign $0.936 - 0.349i$
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  + (−1.35 + 0.397i)2-s + (−0.963 + 0.556i)3-s + (1.68 − 1.07i)4-s + (1.08 − 1.13i)6-s + (−1.26 − 2.32i)7-s + (−1.85 + 2.13i)8-s + (−0.880 + 1.52i)9-s + (1.48 − 0.856i)11-s + (−1.02 + 1.97i)12-s − 2.45·13-s + (2.63 + 2.65i)14-s + (1.67 − 3.63i)16-s + (3.10 + 5.38i)17-s + (0.589 − 2.42i)18-s + (0.108 − 0.187i)19-s + ⋯
L(s)  = 1  + (−0.959 + 0.280i)2-s + (−0.556 + 0.321i)3-s + (0.842 − 0.539i)4-s + (0.443 − 0.464i)6-s + (−0.477 − 0.878i)7-s + (−0.656 + 0.753i)8-s + (−0.293 + 0.508i)9-s + (0.447 − 0.258i)11-s + (−0.295 + 0.570i)12-s − 0.682·13-s + (0.705 + 0.708i)14-s + (0.418 − 0.908i)16-s + (0.754 + 1.30i)17-s + (0.138 − 0.570i)18-s + (0.0248 − 0.0430i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.936 - 0.349i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693132 + 0.125180i\)
\(L(\frac12)\) \(\approx\) \(0.693132 + 0.125180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.397i)T \)
5 \( 1 \)
7 \( 1 + (1.26 + 2.32i)T \)
good3 \( 1 + (0.963 - 0.556i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.48 + 0.856i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.45T + 13T^{2} \)
17 \( 1 + (-3.10 - 5.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.108 + 0.187i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.28 + 5.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.47T + 29T^{2} \)
31 \( 1 + (-0.0819 - 0.141i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.66 - 3.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.34iT - 41T^{2} \)
43 \( 1 + 1.89T + 43T^{2} \)
47 \( 1 + (-10.1 - 5.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.2 + 6.51i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.14 - 3.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.06 - 3.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.58 + 4.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.04iT - 71T^{2} \)
73 \( 1 + (-3.80 - 6.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.8 - 7.97i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.47iT - 83T^{2} \)
89 \( 1 + (-1.54 - 0.891i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−10.42052273165358980927622780645, −9.854454994094339650847074562560, −8.744593013519594226820738890598, −7.957384661806030102152616930128, −7.01426711620698961399858301698, −6.21867244152112798209228323676, −5.30428219313931578925822143284, −4.05418808498489232453435909880, −2.56320102436874245508629780728, −0.827943664946494707091430534887, 0.841512437501465387900651832934, 2.46574093921170156866908309091, 3.44046652699902239470526708507, 5.22310601155901160949385323742, 6.14102455516992340543422200511, 7.00081624047844218620725311074, 7.73209556681732135498750851092, 9.090165349778191278521531149407, 9.339305126080584329208349244798, 10.24384287409908107058872130376

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