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Properties

Label 1-1157-1157.567-r0-0-0
Degree $1$
Conductor $1157$
Sign $-0.226 - 0.974i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
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  + (−0.281 − 0.959i)2-s + (0.977 − 0.212i)3-s + (−0.841 + 0.540i)4-s + (0.654 − 0.755i)5-s + (−0.479 − 0.877i)6-s + (0.997 − 0.0713i)7-s + (0.755 + 0.654i)8-s + (0.909 − 0.415i)9-s + (−0.909 − 0.415i)10-s + (0.755 − 0.654i)11-s + (−0.707 + 0.707i)12-s + (−0.349 − 0.936i)14-s + (0.479 − 0.877i)15-s + (0.415 − 0.909i)16-s + (−0.281 + 0.959i)17-s + (−0.654 − 0.755i)18-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)2-s + (0.977 − 0.212i)3-s + (−0.841 + 0.540i)4-s + (0.654 − 0.755i)5-s + (−0.479 − 0.877i)6-s + (0.997 − 0.0713i)7-s + (0.755 + 0.654i)8-s + (0.909 − 0.415i)9-s + (−0.909 − 0.415i)10-s + (0.755 − 0.654i)11-s + (−0.707 + 0.707i)12-s + (−0.349 − 0.936i)14-s + (0.479 − 0.877i)15-s + (0.415 − 0.909i)16-s + (−0.281 + 0.959i)17-s + (−0.654 − 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.226 - 0.974i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.226 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.529396096 - 1.924995162i\)
\(L(\frac12)\) \(\approx\) \(1.529396096 - 1.924995162i\)
\(L(1)\) \(\approx\) \(1.282761153 - 0.9283437222i\)
\(L(1)\) \(\approx\) \(1.282761153 - 0.9283437222i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.281 - 0.959i)T \)
3 \( 1 + (0.977 - 0.212i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (0.997 - 0.0713i)T \)
11 \( 1 + (0.755 - 0.654i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (0.936 + 0.349i)T \)
23 \( 1 + (-0.936 - 0.349i)T \)
29 \( 1 + (0.997 - 0.0713i)T \)
31 \( 1 + (0.349 + 0.936i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-0.977 - 0.212i)T \)
43 \( 1 + (-0.997 - 0.0713i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (-0.540 + 0.841i)T \)
59 \( 1 + (-0.212 + 0.977i)T \)
61 \( 1 + (-0.599 - 0.800i)T \)
67 \( 1 + (-0.540 + 0.841i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (-0.909 - 0.415i)T \)
83 \( 1 + (0.479 + 0.877i)T \)
97 \( 1 + (-0.755 - 0.654i)T \)
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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

−21.67132170205123303405683946289, −20.627818878636476692008476617347, −19.93835487277948621275903147171, −19.03190485729035031910718360622, −18.100061624758552867852293922119, −17.88189765232483301451585045112, −16.93072525794411533785230729040, −15.8248331392304201445401759110, −15.2118775569246941358878094750, −14.53937243947124927670735114807, −13.83491051799396954798356947469, −13.64274519108004196845448188397, −12.133662454398623542151158295, −11.03784750362483857667665475355, −9.97702340225407342597944756964, −9.54726727792586314072342825324, −8.733147026183951228604019619584, −7.78556704226821354018858091769, −7.20027105995001861199112450354, −6.39949982246891805565023810511, −5.197611259179025368577222928198, −4.52440861913541911807128114116, −3.44623552262290535278021735857, −2.21254030643235009293165811676, −1.4050423912251302227424559756, 1.2520384517073005043237544729, 1.5318533149195842770416410031, 2.59432410491725852935944690540, 3.67137676567356426428010449169, 4.41633984248507045686955747452, 5.353095075572043252165178918639, 6.63501193151530333753667659552, 7.99405063558207980225388326602, 8.460127921320582289740011211304, 8.98813143763789381254798036387, 9.98585280042999279183297379441, 10.5682806941562097259199013514, 11.93222727735455605628178114930, 12.2100697428804326114175426525, 13.37796866327865116182200920104, 13.93999174911656179926346446511, 14.33297247417446959260652393761, 15.58881668891695453896184256919, 16.71923028929752208224497662257, 17.40402098524910740462906579984, 18.13114637702967567423948067009, 18.81849926385528783485843189235, 19.85965871163467655301240462493, 20.158544925850298312399310200807, 20.92622635923990516174152867145

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