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Properties

Label 1-356-356.307-r1-0-0
Degree $1$
Conductor $356$
Sign $0.382 - 0.924i$
Analytic cond. $38.2575$
Root an. cond. $38.2575$
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  + (0.540 − 0.841i)3-s + (0.654 − 0.755i)5-s + (0.755 + 0.654i)7-s + (−0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (0.540 − 0.841i)13-s + (−0.281 − 0.959i)15-s + (0.959 + 0.281i)17-s + (−0.909 + 0.415i)19-s + (0.959 − 0.281i)21-s + (0.909 − 0.415i)23-s + (−0.142 − 0.989i)25-s + (−0.989 − 0.142i)27-s + (0.755 + 0.654i)29-s + (0.909 + 0.415i)31-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)3-s + (0.654 − 0.755i)5-s + (0.755 + 0.654i)7-s + (−0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (0.540 − 0.841i)13-s + (−0.281 − 0.959i)15-s + (0.959 + 0.281i)17-s + (−0.909 + 0.415i)19-s + (0.959 − 0.281i)21-s + (0.909 − 0.415i)23-s + (−0.142 − 0.989i)25-s + (−0.989 − 0.142i)27-s + (0.755 + 0.654i)29-s + (0.909 + 0.415i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(356\)    =    \(2^{2} \cdot 89\)
Sign: $0.382 - 0.924i$
Analytic conductor: \(38.2575\)
Root analytic conductor: \(38.2575\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{356} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 356,\ (1:\ ),\ 0.382 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.784317512 - 1.861874750i\)
\(L(\frac12)\) \(\approx\) \(2.784317512 - 1.861874750i\)
\(L(1)\) \(\approx\) \(1.607829443 - 0.6018863387i\)
\(L(1)\) \(\approx\) \(1.607829443 - 0.6018863387i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.540 - 0.841i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (0.755 + 0.654i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
13 \( 1 + (0.540 - 0.841i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (-0.909 + 0.415i)T \)
23 \( 1 + (0.909 - 0.415i)T \)
29 \( 1 + (0.755 + 0.654i)T \)
31 \( 1 + (0.909 + 0.415i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.540 + 0.841i)T \)
43 \( 1 + (-0.755 + 0.654i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (0.540 + 0.841i)T \)
61 \( 1 + (-0.989 - 0.142i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−24.99340021976546998475998155574, −23.77986106641578423885429909176, −22.86320801846958400949317956280, −21.809974983413809556677170466102, −21.21503484656616325438263383215, −20.64174881392962452379193080810, −19.2813991480308968671491444371, −18.79503383376111585017567090955, −17.2611710177745168013576343120, −16.91936503853351958049505715318, −15.64320389060398213467264827669, −14.68232807806979649775554741064, −13.97956291359383006509847630591, −13.534254878255472962228105611069, −11.606599372220088464630942074516, −10.92588495121082836778577638589, −10.10109438043260506256645942809, −9.12485262550869937986579510687, −8.21900472978123675257883047570, −7.001755916702002942334962325049, −5.91179627038156304807760593151, −4.65430178682754192307037053352, −3.69864874623790134346209024088, −2.640520744097472775400945140823, −1.30077764564805950610948639599, 1.057835031530549426878562436534, 1.78144142730266778216411002226, 2.97375273861206254432816402555, 4.52062554328647897330197650113, 5.65963033472625095222613875248, 6.542081367160010474821884491901, 7.92102272451637711609808977866, 8.56382390334308354254177022881, 9.39377243023237325315272865149, 10.63595752889100657321067354939, 12.16041875847362502851547145718, 12.48338769335847710260968986472, 13.49375703137437476635811337233, 14.5396974596473248911317094360, 15.065352217516550102878707139500, 16.543387999375885294316244890, 17.58233479185256535183345886722, 18.002732635171998592777622975018, 19.09823382693235417820897808299, 20.01149492059500935709526910787, 20.90645817122711624815039877719, 21.380496480149090286441118643107, 22.87144460239001666339939383646, 23.61212989504393007874404426976, 24.71501082004900674935700966428

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