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Properties

Label 2-700-100.23-c1-0-24
Degree $2$
Conductor $700$
Sign $-0.0808 - 0.996i$
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  + (−1.21 + 0.724i)2-s + (0.268 + 0.527i)3-s + (0.949 − 1.75i)4-s + (0.994 + 2.00i)5-s + (−0.708 − 0.446i)6-s + (0.707 + 0.707i)7-s + (0.121 + 2.82i)8-s + (1.55 − 2.14i)9-s + (−2.65 − 1.71i)10-s + (−1.31 − 1.81i)11-s + (1.18 + 0.0280i)12-s + (0.761 + 4.80i)13-s + (−1.37 − 0.346i)14-s + (−0.789 + 1.06i)15-s + (−2.19 − 3.34i)16-s + (3.70 + 1.88i)17-s + ⋯
L(s)  = 1  + (−0.858 + 0.512i)2-s + (0.155 + 0.304i)3-s + (0.474 − 0.879i)4-s + (0.444 + 0.895i)5-s + (−0.289 − 0.182i)6-s + (0.267 + 0.267i)7-s + (0.0429 + 0.999i)8-s + (0.519 − 0.714i)9-s + (−0.840 − 0.541i)10-s + (−0.397 − 0.546i)11-s + (0.341 + 0.00810i)12-s + (0.211 + 1.33i)13-s + (−0.366 − 0.0925i)14-s + (−0.203 + 0.274i)15-s + (−0.548 − 0.835i)16-s + (0.897 + 0.457i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0808 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0808 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.0808 - 0.996i$
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: \((2,\ 700,\ (\ :1/2),\ -0.0808 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.826664 + 0.896422i\)
\(L(\frac12)\) \(\approx\) \(0.826664 + 0.896422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 - 0.724i)T \)
5 \( 1 + (-0.994 - 2.00i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.268 - 0.527i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (1.31 + 1.81i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.761 - 4.80i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-3.70 - 1.88i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.35 - 4.16i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.13 + 7.19i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (4.52 + 1.46i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.25 + 1.05i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.224 + 0.0355i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-5.11 - 3.71i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (7.56 - 7.56i)T - 43iT^{2} \)
47 \( 1 + (-0.175 + 0.0892i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-11.1 + 5.68i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-2.11 - 1.53i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (9.73 - 7.07i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (6.74 - 13.2i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-2.08 - 0.678i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.792 + 0.125i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (1.50 - 4.61i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (14.2 + 7.25i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-1.36 - 1.87i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.59 - 9.00i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.30531458645071299090585470178, −9.879607890371890335821704549971, −8.986175680944164530854966699181, −8.166223809723585149025388062794, −7.16197161344227193371848022920, −6.34668341242699086279735229628, −5.69404632149518916914672277721, −4.21943873969095941197294634864, −2.84464998337488220400838250766, −1.50560962929404293082618517287, 0.932928959550560010324384569431, 2.01706478401976378702339903986, 3.29099494641502638869084260864, 4.75992868603039691328531073098, 5.60783676914174853658238364681, 7.29286138836584580493325796267, 7.63370992061142185585089635708, 8.530027014805005841357776979739, 9.459674700758981726123433962164, 10.14360304737797835584291131199

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