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Properties

Label 1-2312-2312.67-r1-0-0
Degree $1$
Conductor $2312$
Sign $0.709 + 0.705i$
Analytic cond. $248.458$
Root an. cond. $248.458$
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  + (−0.445 − 0.895i)3-s + (−0.850 + 0.526i)5-s + (−0.602 + 0.798i)7-s + (−0.602 + 0.798i)9-s + (−0.932 − 0.361i)11-s + (0.850 + 0.526i)13-s + (0.850 + 0.526i)15-s + (−0.982 + 0.183i)19-s + (0.982 + 0.183i)21-s + (−0.602 + 0.798i)23-s + (0.445 − 0.895i)25-s + (0.982 + 0.183i)27-s + (0.932 + 0.361i)29-s + (−0.850 − 0.526i)31-s + (0.0922 + 0.995i)33-s + ⋯
L(s)  = 1  + (−0.445 − 0.895i)3-s + (−0.850 + 0.526i)5-s + (−0.602 + 0.798i)7-s + (−0.602 + 0.798i)9-s + (−0.932 − 0.361i)11-s + (0.850 + 0.526i)13-s + (0.850 + 0.526i)15-s + (−0.982 + 0.183i)19-s + (0.982 + 0.183i)21-s + (−0.602 + 0.798i)23-s + (0.445 − 0.895i)25-s + (0.982 + 0.183i)27-s + (0.932 + 0.361i)29-s + (−0.850 − 0.526i)31-s + (0.0922 + 0.995i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.709 + 0.705i$
Analytic conductor: \(248.458\)
Root analytic conductor: \(248.458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2312,\ (1:\ ),\ 0.709 + 0.705i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4839645595 + 0.1996946663i\)
\(L(\frac12)\) \(\approx\) \(0.4839645595 + 0.1996946663i\)
\(L(1)\) \(\approx\) \(0.5985982819 - 0.04412959871i\)
\(L(1)\) \(\approx\) \(0.5985982819 - 0.04412959871i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.445 - 0.895i)T \)
5 \( 1 + (-0.850 + 0.526i)T \)
7 \( 1 + (-0.602 + 0.798i)T \)
11 \( 1 + (-0.932 - 0.361i)T \)
13 \( 1 + (0.850 + 0.526i)T \)
19 \( 1 + (-0.982 + 0.183i)T \)
23 \( 1 + (-0.602 + 0.798i)T \)
29 \( 1 + (0.932 + 0.361i)T \)
31 \( 1 + (-0.850 - 0.526i)T \)
37 \( 1 + (0.0922 - 0.995i)T \)
41 \( 1 + (-0.445 - 0.895i)T \)
43 \( 1 + (0.739 - 0.673i)T \)
47 \( 1 + (0.602 + 0.798i)T \)
53 \( 1 + (0.602 - 0.798i)T \)
59 \( 1 + (-0.273 + 0.961i)T \)
61 \( 1 + (-0.273 - 0.961i)T \)
67 \( 1 + (-0.982 + 0.183i)T \)
71 \( 1 + (-0.602 + 0.798i)T \)
73 \( 1 + (-0.739 - 0.673i)T \)
79 \( 1 + (-0.982 + 0.183i)T \)
83 \( 1 + (0.445 - 0.895i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (0.602 - 0.798i)T \)
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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

−19.638615271216840067526204155876, −18.57549632568941850111327067369, −17.85740064319673976308593213827, −16.932799484579031592324042036110, −16.45695798150078264897354108752, −15.76139810243640919818485654675, −15.37897755707897831175051817179, −14.500327865868919111266046210261, −13.40365440452247261918599761883, −12.79872894353787940699271206704, −12.08881436407632955734870015593, −11.186969465883795388531926863445, −10.445847273745261148416773867545, −10.16483319142585727375410701278, −8.98944560660569107191369428456, −8.3745147726313326916634898140, −7.57093323826740607579122237987, −6.56202076160385754809332688413, −5.83736245482586419353903304773, −4.753432264317511208568453556685, −4.33648968078736664955114733351, −3.52582054718068427564136618076, −2.750380981400858309066270686217, −1.097756905094231943329663172893, −0.22662638889130006051296521864, 0.42918041364881433183078203782, 1.81329674995487334544831954461, 2.5848250672681278241133645557, 3.42657865342367033640885938518, 4.36751201651800967140109200858, 5.67103636651194366514124416266, 5.98683053406181444421199858530, 6.93838136039942037044903412783, 7.55538010836854982389743571843, 8.4258492389026215628119121978, 8.93254559083096019287455698074, 10.31792540772387482013399821494, 10.900151225264285843668130628503, 11.60585088059730479557382319748, 12.28420565540173733470037575554, 12.88185851438456278199039496642, 13.65173890339723735584382859047, 14.42223249981383947147424464383, 15.408994676292258708056950206001, 15.98294484927921015662588447100, 16.491974688577255335950598917516, 17.64474018291441522511435390109, 18.271423219580862729163607928011, 18.889831082976084399014649719888, 19.20056529537718491072154074987

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